Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1908.06798

A New Fast Weighted All-pairs Shortest Path Search Algorithm Based on Pruning by Shortest Path Trees

Yasuo Yamane·Kenichi Kobayashi
View original source ↗Read source document ↗Export metadata ↓

Abstract

Recently we submitted a paper, whose title is A New Fast Unweighted All-pairs Shortest Path Search Algorithm Based on Pruning by Shortest Path Trees, to arXiv. This is related to unweighted graphs. This paper also presents a new fast all-pairs shortest path algorithm for weighted graph based on the same idea. In Dijkstra algorithm which is said to be fast in weighted graphs, the average number of accesses to adjacent vertices (expressed by {\alpha}) is about equal to the average degree of the graph. On the other hand, our algorithm utilizes the shortest path trees of adjacent vertices of each source vertex in the same manner as the algorithm for unweighted graphs, and reduce {\alpha} drastically in comparison with Dijkstra algorithm. Roughly speaking {\alpha} is reduced to the value close to 1, because the average degree of a tree is about 2, and one is used to come in and the other is used to go out, although that does not hold true when the depth of the short path trees is small. In case of weighted graphs, a problem which does not occur in unweighted graphs occurs. It is waiting for the generation of the shortest path tree of an adjacent vertex. Therefore, it is possible that a deadlock occurs. We prove our algorithm is deadlock-free. We compared our algorithm with Dijkstra and Peng algorithms. On Dijkstra algorithm ours outperforms it on speed and {\alpha} except that Dijkstra algorithm slightly outperforms ours or they are almost the same on CPU time in sparse scale-free graphs. The result on Peng algorithm is as follows: In speed and {\alpha}, ours outperforms Peng algorithm in hypercube-shaped and dense scale-free graphs, but conversely Peng algorithm outperforms ours in sparse scale-free graphs.

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Yasuo Yamane, Kenichi Kobayashi. 2019-08-19. A New Fast Weighted All-pairs Shortest Path Search Algorithm Based on Pruning by Shortest Path Trees. https://arxiv.org/abs/1908.06798

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗